Questions: (b) Σ from n=2 to ∞ of (-1)^n * 5/ln(n)

Transcript text: (b) $\sum_{n=2}^{\infty}(-1)^{n} \frac{5}{\ln (n)}$

Solution

Solution Steps

To determine the convergence of the series $\sum_{n=2}^{\infty}(-1)^{n} \frac{5}{\ln (n)}$, we can use the Alternating Series Test (Leibniz's Test). This test states that an alternating series $\sum (-1)^n a_n$ converges if the sequence $a_n$ is monotonically decreasing and $\lim_{n \to \infty} a_n = 0$.

Solution Approach

Identify the sequence $a_n = \frac{5}{\ln(n)}$.
Check if $a_n$ is monotonically decreasing.
Check if $\lim_{n \to \infty} a_n = 0$.

Step 1: Identify the Sequence

We define the sequence $ a_n = \frac{5}{\ln(n)} $ for $ n \geq 2 $.

Step 2: Check Monotonicity

We found that the sequence $ a_n $ is monotonically decreasing for $ n \geq 2 $. This means that for all $ n $, $ a_n > a_{n+1} $.

Step 3: Evaluate the Limit

Next, we evaluate the limit of the sequence as $ n $ approaches infinity: \[ \lim_{n \to \infty} a_n = \lim_{n \to \infty} \frac{5}{\ln(n)} = 0.3619 \] Since $ \lim_{n \to \infty} a_n $ is approximately $ 0.3619 $, which is greater than $ 0 $, we conclude that the limit does not approach $ 0 $.

Step 4: Conclusion

Since the sequence $ a_n $ is monotonically decreasing and $ \lim_{n \to \infty} a_n \neq 0 $, the series $ \sum_{n=2}^{\infty} (-1)^{n} \frac{5}{\ln(n)} $ does not converge.

Final Answer

The series diverges, so we conclude that the answer is \$\boxed{\text{diverges}}\$.

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